Journal articles: 'Crystalline cohomology'

1

Crew, Richard. "Specialization of crystalline cohomology." Duke Mathematical Journal 53, no. 3 (September 1986): 749–57. http://dx.doi.org/10.1215/s0012-7094-86-05340-8.

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Nizio?, Wies?awa. "Cohomology of crystalline representations." Duke Mathematical Journal 71, no. 3 (September 1993): 747–91. http://dx.doi.org/10.1215/s0012-7094-93-07128-1.

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Luu, Martin T. "Crystalline cohomology of superschemes." Journal of Geometry and Physics 121 (November 2017): 83–92. http://dx.doi.org/10.1016/j.geomphys.2017.07.005.

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Vonk, Jan. "Crystalline Cohomology of Towers of Curves." International Mathematics Research Notices 2020, no. 21 (September 7, 2018): 7454–88. http://dx.doi.org/10.1093/imrn/rny213.

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Abstract We investigate the geometry of finite maps and correspondences between curves, and construct canonical trace and pullback maps between Hyodo–Kato integral structures on de Rham cohomology of curves, which are functorial for finite morphisms of the generic fibres. This leads to a crystalline version of the étale cohomology of towers of modular curves considered by Hida and Ohta, whose ordinary part satisfies $\Lambda $-adic control and Eichler–Shimura theorems.

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Grosse-Klönne, Elmar. "Equivariant crystalline cohomology and base change." Proceedings of the American Mathematical Society 135, no. 05 (May 1, 2007): 1249–54. http://dx.doi.org/10.1090/s0002-9939-06-08634-5.

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Cais, Bryden, and Tong Liu. "Breuil–Kisin modules via crystalline cohomology." Transactions of the American Mathematical Society 371, no. 2 (September 20, 2018): 1199–230. http://dx.doi.org/10.1090/tran/7280.

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7

Feigin, B. L., and B. L. Tsygan. "Additive K-theory and crystalline cohomology." Functional Analysis and Its Applications 19, no. 2 (1985): 124–32. http://dx.doi.org/10.1007/bf01078391.

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8

Faltings, Gerd, and Bruce W. Jordan. "Crystalline cohomology and GL(2, ℚ)." Israel Journal of Mathematics 90, no. 1-3 (October 1995): 1–66. http://dx.doi.org/10.1007/bf02783205.

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9

Morrow, Matthew. "A Variational Tate Conjecture in crystalline cohomology." Journal of the European Mathematical Society 21, no. 11 (July 19, 2019): 3467–511. http://dx.doi.org/10.4171/jems/907.

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10

Miyatani, Kazuaki. "Finiteness of crystalline cohomology of higher level." Annales de l’institut Fourier 65, no. 3 (2015): 975–1004. http://dx.doi.org/10.5802/aif.2949.

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11

Haastert, Burkhard, and Jens Carsten Jantzen. "Filtrations of symmetric powers via crystalline cohomology." Geometriae Dedicata 37, no. 1 (January 1991): 45–63. http://dx.doi.org/10.1007/bf00150404.

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12

Tsuji, Takeshi. "p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case." Inventiones Mathematicae 137, no. 2 (August 1, 1999): 233–411. http://dx.doi.org/10.1007/s002220050330.

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13

Tsuji, Takeshi. "On nearby cycles and 𝒟-modules of log schemes in characteristic p>0." Compositio Mathematica 146, no. 6 (June 16, 2010): 1552–616. http://dx.doi.org/10.1112/s0010437x10004768.

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AbstractLet K be a complete discrete valuation field of mixed characteristic (0,p) with a perfect residue field k. For a semi-stable scheme over the ring of integers OK of K or, more generally, for a log smooth scheme of semi-stable type over k, we define nearby cycles as a single 𝒟-module endowed with a monodromy ∂logt, whose cohomology should give the log crystalline cohomology. We also explicitly describe the monodromy filtration of the 𝒟-module with respect to the endomorphism ∂logt, and construct a weight spectral sequence for the cohomology of the nearby cycles.

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14

Gillet, Henri, and William Messing. "Cycle classes and Riemann-Roch for crystalline cohomology." Duke Mathematical Journal 55, no. 3 (September 1987): 501–38. http://dx.doi.org/10.1215/s0012-7094-87-05527-x.

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Faltings, Gerd. "Integral crystalline cohomology over very ramified valuation rings." Journal of the American Mathematical Society 12, no. 1 (1999): 117–44. http://dx.doi.org/10.1090/s0894-0347-99-00273-8.

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16

Grosse-Klönne, Elmar. "On the crystalline cohomology of Deligne–Lusztig varieties." Finite Fields and Their Applications 13, no. 4 (November 2007): 896–921. http://dx.doi.org/10.1016/j.ffa.2006.06.001.

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17

Bhatt, Bhargav. "Torsion in the crystalline cohomology of singular varieties." Documenta Mathematica 19 (2014): 673–87. http://dx.doi.org/10.4171/dm/460.

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18

Guo, Haoyang, and Shizhang Li. "Period sheaves via derived de Rham cohomology." Compositio Mathematica 157, no. 11 (October 6, 2021): 2377–406. http://dx.doi.org/10.1112/s0010437x21007545.

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19

Cais, Bryden. "The geometry of Hida families II: -adic -modules and -adic Hodge theory." Compositio Mathematica 154, no. 4 (March 8, 2018): 719–60. http://dx.doi.org/10.1112/s0010437x17007680.

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We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

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Česnavičius, Kęstutis, and Teruhisa Koshikawa. "The -cohomology in the semistable case." Compositio Mathematica 155, no. 11 (September 9, 2019): 2039–128. http://dx.doi.org/10.1112/s0010437x1800790x.

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For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable ${\mathcal{O}}_{K}$-model ${\mathcal{X}}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same ${\mathcal{O}}_{K}$-lattice inside $H_{\text{dR}}^{i}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an $A_{\text{inf}}$-valued cohomology theory of $p$-adic formal, proper, smooth ${\mathcal{O}}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{\text{inf}}$-cohomology to the $p$-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.

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21

Grosse-Klönne, Elmar. "The Cech filtration and monodromy in log crystalline cohomology." Transactions of the American Mathematical Society 359, no. 6 (January 26, 2007): 2945–72. http://dx.doi.org/10.1090/s0002-9947-07-04138-4.

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22

SUZUKI, TAKASHI. "DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES." Nagoya Mathematical Journal 240 (December 19, 2018): 42–149. http://dx.doi.org/10.1017/nmj.2018.46.

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In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

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23

Lauder, Alan G. B. "Ranks of Elliptic Curves Over Function Fields." LMS Journal of Computation and Mathematics 11 (2008): 172–212. http://dx.doi.org/10.1112/s1461157000000565.

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AbstractWe present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology.

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24

OHMORI, Joujuu. "Rationality of certain cuspidal unipotent representations in crystalline cohomology groups." Hokkaido Mathematical Journal 35, no. 3 (August 2006): 547–63. http://dx.doi.org/10.14492/hokmj/1285766415.

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25

Le Stum, Bernard, and Adolfo Quirós. "The Exact Poincaré Lemma in Crystalline Cohomology of Higher Level." Journal of Algebra 240, no. 2 (June 2001): 559–88. http://dx.doi.org/10.1006/jabr.2001.8749.

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26

Lazda, Christopher, and Ambrus Pál. "Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem." Compositio Mathematica 155, no. 5 (May 2019): 1025–45. http://dx.doi.org/10.1112/s0010437x19007164.

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In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for $k$ a perfect field of characteristic $p$ , a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_{p}$ -coefficients.

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27

Tang, Yunqing. "Cycles in the de Rham cohomology of abelian varieties over number fields." Compositio Mathematica 154, no. 4 (March 8, 2018): 850–82. http://dx.doi.org/10.1112/s0010437x17007679.

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In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

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28

Suh, Junecue. "Symmetry and parity in Frobenius action on cohomology." Compositio Mathematica 148, no. 1 (December 8, 2011): 295–303. http://dx.doi.org/10.1112/s0010437x11007056.

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AbstractWe prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

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29

Haastert, Burkhard, and Jens Carsten Jantzen. "Filtrations of the discrete series of SL2(q) via crystalline cohomology." Journal of Algebra 132, no. 1 (July 1990): 77–103. http://dx.doi.org/10.1016/0021-8693(90)90253-k.

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30

Cais, Bryden, and Tong Liu. "Corrigendum to “Breuil–Kisin modules via crystalline cohomology"." Transactions of the American Mathematical Society 373, no. 3 (November 18, 2019): 2251–52. http://dx.doi.org/10.1090/tran/7894.

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31

Joshi, Kirti. "Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence." Canadian Mathematical Bulletin 50, no. 4 (December 1, 2007): 567–78. http://dx.doi.org/10.4153/cmb-2007-054-9.

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AbstractIn this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic p > 0 is Hodge–Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N. Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

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32

Dimitrov, Mladen. "On Ihara’s lemma for Hilbert modular varieties." Compositio Mathematica 145, no. 5 (September 2009): 1114–46. http://dx.doi.org/10.1112/s0010437x09004205.

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AbstractLet ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

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33

Le Stum, Bernard, and Adolfo Quirós. "The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)." Nagoya Mathematical Journal 191 (2008): 79–110. http://dx.doi.org/10.1017/s0027763000025915.

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AbstractWe show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

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34

Langer, Andreas, and Thomas Zink. "Comparison between overconvergent de Rham-Witt and crystalline cohomology for projective and smooth varieties." Mathematische Nachrichten 288, no. 11-12 (April 27, 2015): 1388–93. http://dx.doi.org/10.1002/mana.201400220.

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35

Lodh, Rémi Shankar. "Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case." Annales de l’institut Fourier 61, no. 5 (2011): 1875–942. http://dx.doi.org/10.5802/aif.2661.

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36

Loeffler, David, and Sarah Livia Zerbes. "Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions." International Journal of Number Theory 10, no. 08 (October 29, 2014): 2045–95. http://dx.doi.org/10.1142/s1793042114500699.

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We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

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37

Langer, Andreas, and Thomas Zink. "A comparison of logarithmic overconvergent de Rham–Witt and log-crystalline cohomology for projective smooth varieties with normal crossing divisor." Rendiconti del Seminario Matematico della Università di Padova 137 (2017): 229–35. http://dx.doi.org/10.4171/rsmup/137-13.

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38

Mondal, Shubhodip. "Dieudonné theory via cohomology of classifying stacks." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.77.

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Abstract We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$ ) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.

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39

OLSSON, Martin C. "Crystalline cohomology of algebraic stacks and Hyodo-Kato cohomology." Astérisque, November 6, 2018. http://dx.doi.org/10.24033/ast.753.

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40

Matsumoto, Keiho. "GYSIN TRIANGLES IN THE CATEGORY OF MOTIFS WITH MODULUS." Journal of the Institute of Mathematics of Jussieu, January 6, 2022, 1–24. http://dx.doi.org/10.1017/s1474748021000554.

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Abstract In this article, we study a Gysin triangle in the category of motives with modulus (Theorem 1.2). We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives (Corollary 1.6). The corollary implies that an object in $\operatorname {\mathbf {MDM}^{\operatorname {eff}}}$ decomposes into a p-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.

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41

Kedlaya, Kiran S. "Etale and crystalline companions, I." Épijournal de Géométrie Algébrique Volume 6 (December 2, 2022). http://dx.doi.org/10.46298/epiga.2022.6820.

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Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$. Using the Langlands correspondence for global function fields in both the \'etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any \'etale coefficient object has \'etale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has \'etale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of \'etale coefficient objects; this subject will be pursued in a subsequent paper.

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42

Gregory, Oliver, and Andreas Langer. "Hodge–Witt decomposition of relative crystalline cohomology." Journal of the London Mathematical Society, September 30, 2022. http://dx.doi.org/10.1112/jlms.12679.

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43

Colmez, Pierre, and Wiesława Nizioł. "On the cohomology of 𝑝-adic analytic spaces, I: The basic comparison theorem." Journal of Algebraic Geometry, July 26, 2024. http://dx.doi.org/10.1090/jag/835.

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The purpose of this paper is to prove a basic p p -adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure C C of a p p -adic field: p p -adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over B dR + {\mathbf B}^+_{\operatorname {dR} } ). The key computation is the passage from absolute crystalline cohomology to Hyodo–Kato cohomology and the construction of the related Hyodo–Kato isomorphism. We also “geometrize” our comparison theorem by turning p p -adic pro-étale and syntomic cohomologies into sheaves on the category P e r f C {\mathrm {Perf}}_C of perfectoid spaces over C C and the period morphisms into maps between such sheaves (this geometrization will be crucial in our study of the C s t C_{\mathrm {st}} -conjecture in the sequel to this paper and in the formulation of duality for geometric p p -adic pro-étale cohomology).

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44

Trihan, Fabien, and David Vauclair. "A Comparison Theorem for Semi-Abelian Schemes over a Smooth Curve." Memoirs of the American Mathematical Society 299, no. 1495 (July 2024). http://dx.doi.org/10.1090/memo/1495.

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We compare flat cohomology to crystalline syntomic complexes with coefficients in two cases: (1) p p -divisible groups over a separated F p \mathbb {F}_p -scheme with local finite p p -bases, (2) semi-abelian schemes over a separated irreducible smooth curve.

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45

Yao, Zijian. "LOGARITHMIC DE RHAM–WITT COMPLEXES VIA THE DÉCALAGE OPERATOR." Journal of the Institute of Mathematics of Jussieu, August 26, 2021, 1–64. http://dx.doi.org/10.1017/s1474748021000402.

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Abstract We provide a new formalism of de Rham–Witt complexes in the logarithmic setting. This construction generalises a result of Bhatt–Lurie–Mathew and agrees with those of Hyodo–Kato and Matsuue for log-smooth schemes of log-Cartier type. We then use our construction to study the monodromy action and slopes of Frobenius on log crystalline cohomology.

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46

Song, Hao, Charles Zhaoxi Xiong, and Sheng-Jie Huang. "Bosonic crystalline symmetry protected topological phases beyond the group cohomology proposal." Physical Review B 101, no. 16 (April 22, 2020). http://dx.doi.org/10.1103/physrevb.101.165129.

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47

JOSHI, KIRTI. "ON VARIETIES WITH TRIVIAL TANGENT BUNDLE IN CHARACTERISTIC 0$" height="12pt">." Nagoya Mathematical Journal, June 26, 2019, 1–17. http://dx.doi.org/10.1017/nmj.2019.19.

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In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$ . Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

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48

Moon, Yong Suk. "Strongly divisible lattices and crystalline cohomology in the imperfect residue field case." Selecta Mathematica 30, no. 1 (December 27, 2023). http://dx.doi.org/10.1007/s00029-023-00899-y.

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49

Sheinbaum, Daniel, and Omar Antolín Camarena. "Crystallographic interacting topological phases and equivariant cohomology: to assume or not to assume." Journal of High Energy Physics 2021, no. 7 (July 2021). http://dx.doi.org/10.1007/jhep07(2021)139.

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Abstract For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the P-equivariant cohomology of a d-dimensional torus gives rise to different interacting phases, where P denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.

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50

LE, DANIEL, BAO V. LE HUNG, BRANDON LEVIN, and STEFANO MORRA. "SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE." Forum of Mathematics, Pi 8 (2020). http://dx.doi.org/10.1017/fmp.2020.1.

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We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$ -dimensional Galois representations’, Duke Math. J. 149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$ .

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Journal articles on the topic 'Crystalline cohomology'

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